An Asymptotic Study of Blow up Multiplicity in Fourth Order Parabolic Partial Differential Equations

Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we predict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two spatial dimensions, the singular perturbation theory reveals that the set of possible singularity points depends subtly on the geometry of the domain and the equation parameters. In three spatial dimensions, preliminary numerical simulations indicate that the multiplicity of singularities can be even more complex. For the aforementioned scenarios, the analysis highlights the dichotomy of behaviors exhibited between the second and fourth order cases.